| L(s) = 1 | − 2.53·2-s + 3-s + 4.41·4-s − 2·5-s − 2.53·6-s + 0.532·7-s − 6.10·8-s + 9-s + 5.06·10-s − 1.81·11-s + 4.41·12-s − 3.75·13-s − 1.34·14-s − 2·15-s + 6.63·16-s − 1.30·17-s − 2.53·18-s − 0.652·19-s − 8.82·20-s + 0.532·21-s + 4.59·22-s − 1.14·23-s − 6.10·24-s − 25-s + 9.51·26-s + 27-s + 2.34·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.577·3-s + 2.20·4-s − 0.894·5-s − 1.03·6-s + 0.201·7-s − 2.15·8-s + 0.333·9-s + 1.60·10-s − 0.547·11-s + 1.27·12-s − 1.04·13-s − 0.360·14-s − 0.516·15-s + 1.65·16-s − 0.316·17-s − 0.596·18-s − 0.149·19-s − 1.97·20-s + 0.116·21-s + 0.979·22-s − 0.238·23-s − 1.24·24-s − 0.200·25-s + 1.86·26-s + 0.192·27-s + 0.443·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 0.532T + 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 0.652T + 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 - 0.411T + 37T^{2} \) |
| 41 | \( 1 + 4.50T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 + 7.75T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 7.84T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 1.30T + 79T^{2} \) |
| 83 | \( 1 - 6.04T + 83T^{2} \) |
| 89 | \( 1 + 1.68T + 89T^{2} \) |
| 97 | \( 1 - 9.27T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.34638647476071480824940754972, −9.698100674742979211120368701628, −8.728313808317447280463629642004, −7.995896000741799387331699281421, −7.50024369063529895266256727961, −6.53882681663123716704627366127, −4.75808287635147761068745418353, −3.15074197015351106435602717250, −1.91891018253696854751520250607, 0,
1.91891018253696854751520250607, 3.15074197015351106435602717250, 4.75808287635147761068745418353, 6.53882681663123716704627366127, 7.50024369063529895266256727961, 7.995896000741799387331699281421, 8.728313808317447280463629642004, 9.698100674742979211120368701628, 10.34638647476071480824940754972
